Well, it just comes down to whether [itex]\mathbb{E}\left[Var(X_1 + X_2 +...)\right]=\mathbb{E}\left[X_1\right]+\mathbb{E}\left[X_2\right]+...[/itex], right? Since, by plugging in the definition of variance, you can change the variance of the sum into the expectation of a sum. And that comes down to whether you're allowed to move a limit (since that's what the infinite sum is, formally) out of a Lebesque integral. For that you need a convergence theorem, e.g. dominated convergence or something like that, which is going to place some limits on the Xi, but they'll not be too severe. I think (I'm too lazy to try to prove this), that if the sum on the right converges, that's probably enough to prove the convergence of the integral on the left to the same value.